Could also reccomend Rogue with Scout subclass. You get that great woodsy feel (plus automatic expertise in Survival and Nature) with a ton of great skill choices. You won't be putting down a storm of arrows but the ones you do put out there will hit like a truck. Use stealth and Steady Aim to get advantage when needed and you can easily put together a Ghost of the Forest vibe. Strike from nowhere, vanish into the mist and shadows. Pick up Skulker feat along the way, maybe Fighting Initiate for Archery style or take a dip into Fighter for it (as well as Second Wind). 

Unless you KNOW that the campaign is going to go to 20, do not plan on anything that will happen at 20.

I'm not certain but I think that most campaigns end around 15 or so?

There are also more things to consider than just plain combat effectiveness.

Most campaigns either fizzle early (by L6) or don’t go past L15 unless the DM is running things in a long running world. Even then by L15 most characters retire  either to be put away except for high level  one shots or to become NPCs in a shared world. I typically don’t look past L11 when I’m comparing things for my own use. Leaving aside multiclass builds the two subclasses I’ve heard discussed are the battle master and the Samuri of them I think the battlemaster is to be preferred as it has more use of its maneuvers (5/short rest) vs the Samuri’s advantage on attacks 3 turns per long rest (granted at L11 that is actually 9 shots total but it’s still only 3 turns/LR).  Have you considered the arcane archer? You only get 2 uses of its arcane shot per short rest but that typically works out to 6 uses per long rest/day. The other possibility would be the horizon walker ranger. Up thru L10 the ranger gets as many shots as the fighter and the L11 ability of the horizon walker gives it a third shot along with two teleports to empty spaces each round. That can get/keep you out of trouble/melee very easily as teleports are not subject to AoO. The big advantage fighters really get is the extra ASI at L6. As a wood elf you are actually able to get two racial feats - elven accuracy and wood elf magic. You might speak to the DM about getting both some will allow it. In addition to the racial feat(s) as a fighter you get 4 ASIs/feats as a ranger you get 3. You said you need two to max stats that leaves 1 for sharpshooter and, if a fighter, 1 for either CBE, Piercer or possible fighting initiate to take a second fighting style to cover melee situations ( defense’s +1AC comes to mind)

+1 for scrapping any thoughts about level 20 unless the campaign is running at level 20!

character creation is one of those things which lots of people do differently. Personally, I make a concept of a character (for example; a guy who's got this one weapon which he's trained every day with, and doesn't wear armour because "wearing armour is admitting the possibility of being hit"). Then I browse the classes for something which fits - in this case, I'd look over the Monk, Fighter and Barbarian classes, and then I'd make something from there, bending my original plan to fit the rules and reflavourign the rules to fit the plan.

It's also worth noting that who your character will be at level 20, assuming you're starting in teir 1 or 2, is going to be defined by the game, not by your plan from the start, like as not. You might want to make an amazing archer, but after 5 levels of playing said archer, you might say "you know what? I'm going to branch into Warlock for some magical goodness!".

At any rate, I recommend deciding what your character is like at the start of the campaign (EG, a skilled archer) and building that, rather than focussing on what they will be at the end of the campaign (a crazily skilled archer) and worrying over which classes will get you there!